A sequence of remarkable Poincaré inequalities for Dunkl operators.

We consider the Dunkl differential-difference operator associated with a reflection group W and a nonnegative parameter function k. We exhibit and study a new family of sharp integral inequalities related to the Dunkl heat semigroup generated by the Dunkl-Laplacian . These estimations are a natural extension and refinement of the Poincaré inequality established in [1] and used to derive other applications. [ABSTRACT FROM AUTHOR]